One source close to Mr Cameron said at the weekend: “There is a special place in hell reserved for Boris. We need to get behind Theresa. She’s the grown up.”

Senior party chiefs are expecting half a dozen Tory MPs to stand for the leadership.

Amber Rudd, the Energy and Climate Change secretary, is also being pushed as a candidate who can continue Mr Cameron’s modernisation agenda, while Sajid Javid, the Business secretary, is mulling a leadership bid and is set to make a statement later this wee

Among Brexiteers they include Justice minister Dominic Raab, Treasury minister Andrea Leadsom and Work and Pensions minister Priti Patel.

Other challengers among pro-EU MPs could include Stephen Crabb, the Work and Pensions secretary, Health secretary Jeremy Hunt, and Nicky Morgan, the Education secretary.

Life sciences minister George Freeman has also let it be known that he will use support among the 2010 and 2015 Parliamentary intakes to mount a challenge.

Mrs May has emerged relatively unscathed from the Government’s disastrous EU referendum campaign because she painted herself as a ‘reluctant remainer’.

She made only a handful of interventions during the campaign, including a detailed speech which observers said could have been an argument for either leaving or remaining in the EU into the final page.

Mrs May faces an immediate challenge to win widespread support not least because of her unclubbable icy reputation among MPs in Westminster.

She is also hampered by not having a natural constituency of MPs in the parliamentary party give that she was elected with a relatively small number of Tory MPs at the time of Tony Blair’s Labour landslide in 1997.

Mrs May also famously upset activists in 2002 when as the party’s first female chairman she urged the party to change because it was seen as “the nasty party”.

Mrs May has also held a number of briefs in opposition – including culture, environment, transport, work and pensions and a period as shadow of the house – which gives her more experience than Mr Johnson.

In 2013, Mrs May revealed she has been diagnosed with Type 1 diabetes but insisted it would not affect her political career.

The condition means her body does not produce insulin and she must now inject herself with the hormone at least twice a day for the rest of her life.